The Wolverine opens with Logan shielding Yashida during the bombing of Nagasaki. He's badly burned and regenerates.

The Wolverine opens with Logan shielding Yashida during the bombing of Nagasaki. He's badly burned and regenerates. He has taken cover in a well. That's important. The film shows him surviving the blast and burns in that situation, not standing completely unprotected at the center of the explosion. You don't have to exaggerate it to make the recovery impressive. Later, his injuries stop clearing up the way he expects. Gunshots become a much more serious problem while he's trying to protect Mariko. He finds the mechanical parasite attached to his heart and cuts himself open to remove it. His recovery after removing it gives that weakness a specific cause. This is a case where something is suppressing the power. It wouldn't make sense to use those weakened moments as his normal healing speed, or to pretend that having the power means nothing can interfere with it.